Variable Transformations in combination with Wavelets and ANOVA for high-dimensional approximation

Abstract

We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered-data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions we propose a new extension method. A proper choice of the extension parameter together with the piece-wise polynomial Chui-Wang wavelets extends the functions appropriately. In every case we are able to bound the approximation error with high probability. Additionally, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

Sparse additive function decompositions facing basis transforms

“…The approximation of high-dimensional functions is a classical topic of mathematical analysis with rich real-world applications. Due to new numerical techniques arising from (stochastic) Fourier-and wavelet [7,43,36,37] as well as kernel [40] methods and deep learning approaches, the topic has recently attained increasing attention. For example, including information about the structure of the function class of interest into a deep neural network architecture can improve its approximation quality [3,17,14].…”